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Chapter 1
Chapter 1
Solving Equations and Inequalities
1.1 – Expressions and Formulas
1.1 – Expressions and Formulas
Order of Operations
1.1 – Expressions and Formulas
Order of Operations
Parentheses
1.1 – Expressions and Formulas
Order of Operations
Parentheses
Exponents
1.1 – Expressions and Formulas
Order of Operations
Parentheses
Exponents
Multiplication
1.1 – Expressions and Formulas
Order of Operations
Parentheses
Exponents
Multiplication
Division
1.1 – Expressions and Formulas
Order of Operations
Parentheses
Exponents
Multiplication
Division
Addition
1.1 – Expressions and Formulas
Order of Operations
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents
Multiplication
Division
Addition
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents Excuse
Multiplication
Division
Addition
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents Excuse
Multiplication My
Division
Addition
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents Excuse
Multiplication My
Division Dear
Addition
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents Excuse
Multiplication My
Division Dear
Addition Aunt
Subtraction
1.1 – Expressions and Formulas
Order of Operations
Parentheses Please
Exponents Excuse
Multiplication My
Division Dear
Addition Aunt
Subtraction Sally
Example 1
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 =
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 = [2(6)2 + 3] ÷ 5
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 = [2(6)2 + 3] ÷ 5
[2(36) + 3] ÷ 5
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 = [2(6)2 + 3] ÷ 5
[2(36) + 3] ÷ 5
[72 + 3] ÷ 5
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 = [2(6)2 + 3] ÷ 5
[2(36) + 3] ÷ 5
[72 + 3] ÷ 5
75 ÷ 5
Example 1
Find the value of [2(10 - 4)2 + 3] ÷ 5.
[2(10 - 4)2 + 3] ÷ 5 = [2(6)2 + 3] ÷ 5
[2(36) + 3] ÷ 5
[72 + 3] ÷ 5
75 ÷ 5
15
Example 2
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) =
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5)
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5)
82 – 1.5(8 + 1.5)
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5)
82 – 1.5(8 + 1.5)
82 – 1.5(9.5)
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5)
82 – 1.5(8 + 1.5)
82 – 1.5(9.5)
64 – 1.5(9.5)
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5)
82 – 1.5(8 + 1.5)
82 – 1.5(9.5)
64 – 1.5(9.5)
64 – 14.25
Example 2
Evaluate x2 – y(x + y) if x = 8 and y = 1.5.
x2 – y(x + y) = 82 – 1.5(8 + 1.5) 82 – 1.5(8 + 1.5)
82 – 1.5(9.5) 64 – 1.5(9.5) 64 – 14.25 49.75
Example 3
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
a3 + 2bc =
c2 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
a3 + 2bc = 23 + 2(-4)(-3)
c2 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
a3 + 2bc = 23 + 2(-4)(-3)
c2 – 5 (-3)2 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
a3 + 2bc = 23 + 2(-4)(-3)
c2 – 5 (-3)2 – 5
= 8 + 2(-4)(-3)
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.
c2 – 5
a3 + 2bc = 23 + 2(-4)(-3)
c2 – 5 (-3)2 – 5
= 8 + 2(-4)(-3)
9 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.c2 – 5
a3 + 2bc = 23 + 2(-4)(-3) c2 – 5 (-3)2 – 5
= 8 + 2(-4)(-3) 9 – 5
= 8 + 24 9 – 5
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.c2 – 5
a3 + 2bc = 23 + 2(-4)(-3) c2 – 5 (-3)2 – 5
= 8 + 2(-4)(-3) 9 – 5
= 8 + 24 9 – 5
= 32 4
Example 3
Evaluate a3 + 2bc if a = 2, b = -4, and c = -3.c2 – 5
a3 + 2bc = 23 + 2(-4)(-3) c2 – 5 (-3)2 – 5
= 8 + 2(-4)(-3) 9 – 5
= 8 + 24 9 – 5
= 32 = 8 4
Example 4
Example 4
Find the area of the following trapezoid. 16 in.
10 in.
52 in.
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
A = ½h(b1 + b2)
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in. = h
52 in.
A = ½h(b1 + b2)
Example 4
Find the area of the following trapezoid. 16 in. = b1
A = ½h(b1 + b2)
10 in. = h
52 in.
A = ½h(b1 + b2)
Example 4
Find the area of the following trapezoid. 16 in. = b1
A = ½h(b1 + b2)
10 in. = h
52 in. = b2
A = ½h(b1 + b2)
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
A = ½h(b1 + b2)
= ½10(16 + 52)
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
A = ½h(b1 + b2)
= ½10(16 + 52)
= ½10(68)
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
A = ½h(b1 + b2)
= ½10(16 + 52)
= ½10(68)
= 5(68)
Example 4
Find the area of the following trapezoid. 16 in.
A = ½h(b1 + b2)
10 in.
52 in.
A = ½h(b1 + b2)
= ½10(16 + 52)
= ½10(68)
= 5(68) = 340
1.2 – Properties of Real Numbers
1.2 – Properties of Real Numbers
Real Numbers
1.2 – Properties of Real Numbers
Real Numbers (R)
1.2 – Properties of Real Numbers
Real Numbers (R)
1.2 – Properties of Real Numbers
Real Numbers (R)
Rational
1.2 – Properties of Real Numbers
Real Numbers (R)
Rational (⅓)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
Integers
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
Integers (-6)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
Whole #’s
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
Whole #’s (0)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
Natural #’s
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
Natural #’s (7)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (7)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓) Irrational
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓) Irrational √ 5
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
1.2 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓) (I) Irrational √ 5
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
Real
Rational Irrational Integers
Whole Natural
Example 1
Example 1
Name the sets of numbers to which each apply.
Example 1
Name the sets of numbers to which each apply.
Example 1
Name the sets of numbers to which each apply.
Example 1
Name the sets of numbers to which each apply.
(a) √ 16
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞ - Q
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞ - Q, R
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞ - Q, R
__
(e) 0.45
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞ - Q, R
__
(e) 0.45 - Q
Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
(c) √ 20 - I, R
(d) -⅞ - Q, R
__
(e) 0.45 - Q, R
Properties of Real Numbers
Property Addition Multiplication
Commutative a + b = b + a a·b = b·a
Associative (a+b)+c = a+(b+c) (a·b)·c = a·(b·c)
Identity a+0 = a = 0+a a·1 = a = 1·a
Inverse a+(-a) =0= -a+a a·1 =1= 1·a
a a
Distributive a(b+c)=ab+ac and (b+c)a=ba+ca
Example 2
Example 2
Name the property used in each equation.
Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Commutative Addition
Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Commutative Addition
(b) 3(4x) = (3·4)x
Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Commutative Addition
(b) 3(4x) = (3·4)x
Associative Multiplication
Example 3
What is the additive and multiplicative inverse for -1¾?
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + = 0
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: -1¾
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: -1¾ · = 1
Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: (-1¾)(-4/7) = 1
Example 4
Example 4
Simplify 2(5m+n)+3(2m–4n).
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m +
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n +
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m –
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
10m + 6m + 2n – 12n
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
10m + 6m + 2n – 12n
16m
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m + 2n + 6m – 12n
10m + 6m + 2n – 12n
16m – 10n
